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```616                    ON THE KROBLEM OF RANDOM VIBKATIONS,  AND                  [441
If p = ls s = - -|, and the cosine factors in (38) become
cos (£ TT + r#) cos4 (%TT — las), yielding finally
/5-7T               .,   \             /07T               ,7   \
cos  --T- + rx - 4>las},  cos  -r- - rx - 4te  , V 4                    y            V 4                    /
cos   -   + ra-2to,   cos     - 7*0 -        ,  cos V 4                 /         \4                /          \4
so that there is no constant terra unless r = 4Z, or 21   With these exceptions, the original differentiation under the integral sign is justified.
We fall back upon <j>t itself by putting p = 0, making s = - f. The integral is then finite in all cases (r £ 0), in agreement with Pearson's curve.
Next for n = 5, s = p — 2.
When p = l,s = -~l, and we find that the cosine factors yield a constant term only when r = 31 Pearson's curve does not suggest anything special at r=Bl'} it may be remarked that the integral with p = I is there only logarithmically' infinite.
If n = 5, p = 0, s = - 2; and the integral for <£„ is finite for all values of r.
When n - 6, s = p - 2\$. In this case, whether p = 1, or 0, no question can arise. The integrals are finite for all values of ?-.
A fortiori is this so, when n > 6.
If we suppose p = 2, s = £ (5 - w). Thus n = T makes s = -1, and infinities might occur for special values of r. But if w>"7, s< —f, and infinities are excluded whatever may be the value of r.
Similarly if p = 3, infinities are excluded if n > 9, and so on.
Our discussion has not yet yielded all that could be wished; the subject may be commended to those better versed in pure mathematics. Probably what is required is a better criterion as to the differentiation under the integral sign*.
We may now pass oh to consider what becomes of Kluyver's integral when n is made infinite. As already remarked, Pearson has developed for it a series proceeding by powers of l/n, and it may be convenient to give a version of his derivation, without, however, carrying the process so far.
[* The criterion enunciated on p. 614 appears to have been devised to meet the case when s = 0 and the integral,, though finite, does not converge to a definite value when x = <x>. If, however, ,s < -1, or <0, respectively, according as the cosine factors in (38) do or do not produce a constant term, the integral (38) has been shown to be finite; it is -also convergent; and the integrals obtained by omitting before each successive differentiation the factor to be differentiated, viz.': cos (%ir-rx-lqir) where g<p, are also finite (cf. Todhifnter's Integral Calculus, 1889, Arts. 214, 284). In these circumstances it would appear that-_(38) is itself valid, and that it is unnecessary to consider the integral obtained by "the next following operation" (s =p + f - %ri). It would seem then that the above considerations are sufficient to justify the differentiation by which 0,t is obtained (jp = 0, s- -f), and a fortiori that for <fe (# = 0, s=-2), etc. W..IY.S.]elative index is then much restricted, the
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